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The main result is that, for any projective compact analytic subset Y of dimension q > 0 in a reduced complex space X, there is a neighborhood Ω of Y such that, for any covering space \({\Upsilon\colon\widehat X\to X}\) in which \({\widehat Y\equiv\Upsilon^{-1}(Y)}\) has no noncompact connected analytic subsets of pure dimension q with only compact irreducible components, there exists a C ∞ exhaustion function \({\varphi}\) on \({\widehat X}\) which is strongly q-convex on \({\widehat\Omega=\Upsilon^{-1}(\Omega)}\) outside a uniform neighborhood of the q-dimensional compact irreducible components of \({\widehat Y}\).

Strong q-convexity in uniform neighborhoods of subvarieties in coverings of complex spaces